few_shots: Example:
Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Standard options: 
A. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
B. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
C. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)


Thought: This problem solves a real-world problem involving mixing ingredients for cookies. A student doing this problem would need to reason about equivalent ratios of butter amounts to sugar amounts.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_000972

6
options: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
problem_activity: Activity
In this partner activity, students take turns solving equations involving cube roots by isolating the cube root expression and then cubing each side of the equation. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).
With square roots, students would have to check that their solutions are valid by substituting in the original equation. That isn’t necessary with cube roots because there is only ever one cube root, but it is a good habit to have.
Launch
Arrange students in groups of 2. After students have had time to identify their three most and least difficult equations, ask students to take turns: the first partner identifies an equation from one of their lists and explains why they chose it, while the other listens and works to understand. Then they switch roles. When both partners have explained why they chose one equation from their most difficult list and one from their least difficult list, they move on to the second question.
When both partners have solved the four equations they chose, ask them to take turns again: the first partner explains how they solved one of their problems, while the other listens and works to understand. When both partners agree on the solution, they switch roles. They should continue to switch roles as time allows.
Conversing: MLR2 Collect and Display.
As students work on this activity, listen for and collect the language students use to explain how they solved the equations. Write the students’ words and phrases on a visual display. As students review the visual display, ask them to clarify the meaning of a phrase. For example, a phrase such as, “Add 4 and then raise to the third power,” can be clarified by restating it as, “Add 4 to each side of the equation. Then raise each side of the equation to the third power.” This will help students use the mathematical language necessary to precisely explain how to solve equations involving cube roots.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer collaboration. When students share the equations they selected with a partner, display sentence frames to support conversation such as: “I chose _____ (equation) because . . .”, “I noticed _____ so I chose . . .”, and “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Here are a lot of equations:
\(\sqrt[3]{t+4} = 3\)
\(\text-10 = \text-\sqrt[3]{a}\)
\(\sqrt[3]{3-w} - 4 = 0\)
\(\sqrt[3]{z} + 9 = 0\)
\(\sqrt[3]{r^3 - 19} = 2\)
\(5 - \sqrt[3]{k+1} = \text-1\)
\(\sqrt[3]{p+4} - 2 = 1\)
\(6-\sqrt[3]{b} = 0\)
\(\sqrt[3]{2n} + 3 = \text-5\)
\(4+\sqrt[3]{\text-m} + 4 = 6\)
\(\text-\sqrt[3]{c} = 5\)
\(\sqrt[3]{s-7} + 3 = 0\)
Without solving, identify 3 equations that you think would be the least difficult to solve and 3 equations that you think would be the most difficult to solve. Be prepared to explain your reasoning.
Choose 4 equations and solve them. At least one should be from your “least difficult” list and at least one should be from your “most difficult” list.
Student Response
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Student Facing
Are you ready for more?
All of these equations were equivalent to equations that could be written in the form
\(\sqrt[3]{ax+b}+c=0\)
for some constants
\(a\)
,
\(b\)
, and
\(c\)
. Find a formula that would solve any such equation for
\(x\)
in terms of
\(a\)
,
\(b\)
, and
\(c\)
.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
If students have trouble getting started on their more difficult equations, remind them of strategies they used with square roots, such as isolating the root first. Some students may find it helpful to estimate an answer and then check it, in order to get a general idea of what the answer may be and the calculations involved in finding it. If needed, students can look at the graph of
\(y=\sqrt[3]{x}\)
from the previous activity, or
\(y=x^3\)
from earlier in the lesson.
Students who don’t reach a solution should say where they got stuck and what they tried when they share their work with their partner. They can then try finishing the problem together.
Activity Synthesis
Much discussion takes place between partners. Invite students to share how they solved the equations they chose. Here are some questions for discussion:
“How is the process of solving these equations the same as solving equations that involve square roots? How is it different?” (Solving these equations is different because they require cubing rather than squaring. It is also different because cube roots aren’t always positive, so the extra step to check that a positive number isn’t set equal to a negative number isn’t necessary. Solving is the same because it requires isolating the radical expression and then using exponents to make a new equation without radicals.)
“Describe any difficulties you experienced and how you resolved them.” (I cubed a negative number and forgot that the result would be negative; when I plugged in my answer to check it, I saw that I must have done something wrong, so I went back and fixed it. I wasn’t sure how to get started on one of my difficult equations, but I remembered that I should try to isolate the cube root, and then I could see what to do.)


correct_option_index: 1